Gauss-Seidel iterative techniqueDescriptions of load flow solution techniques can become rather complicated, due more to the notation required for complex arithmetic rather than the basic concepts of the solution method. In the following sections, therefore, the basic techniques are developed by considering their application to a dc circuit. Applications to ac problems are then a natural extension of the dc problem.The Gauss-Seidel solution algorithm, although not the most powerful, is the easiest to understand. The performance of the Gauss-Seidel technique will be illustrated using the direct current circuit shown in Figure-I.Bus 3 is a load bus with specified per unit power. Bus 2 is a generator bus with power specified, and bus 1 is the swing bus with voltage specified. The voltagesV2 and V3 are sought. From these, the branch flows can be calculated.The system equations on an admittance basis are from Equation (6-2).

The terms of the admittance matrix are easily determined from the circuit [2], [3], [4], [7].The off-diagonal term Yijis the negative of the line admittance between bus I and bus j

The diagonal terms are the sum of the admittances of the lines leaving a bus plus the admittance of the bus shunt plus one-half of the charging admittance for each connected line. The Y matrix is very sparse (few nonzero elements), so special matrix techniques are often used to minimize computer storage requirements. From Equation xx,

This is a nonlinear equation in V2.For bus 3, a similar procedure yields

where the negative sign on P3 is from the load sign convention. Equations (6-10) and (6-11) are in a form convenient for the application of the Gauss-Seidel iterative solution technique. The steps in this procedure are as follows:a) Step 1: Assign an estimate of V2 and V3 (for example,V2=V3= 1.0). Note thatV1isfixed.b) Step 2: Compute a new value forV2using the initial estimates forV2andV3[seeEquation (6-10)].c) Step 3: Compute a new value forV3using the initial estimate forV3and the justcomputed value forV2[see Equation (6-11)].d) Step 4: Repeat b) and c) using the latest computed voltagesV2andV3until thesolution is reached. One complete computation ofV2andV3is one iteration.The computed voltages are said to converge when, for each iteration, they come closer andcloser to the actual solution satisfying the network equations. Since the computer timeincreases linearly with the number of iterations, it is necessary to have the computer programmake a check after each iteration and decide whether the last computed voltages aresufficiently close to the true solution or whether further computations are required. Thecriterion specifying the desired accuracy is called the convergence criterion.A reliable convergence criterion is the power mismatch check. Based on the last computedvoltage solution, the sum of the power flows (real and reactive) on all lines connected to thebus and to the bus shunt is compared with the specified bus real and reactive power. Thedifference, which is the power mismatch, is a measure of how close the computed voltagesare to the true solution. The power mismatch tolerance is generally specified in the range of0.01 to 0.0001 p.u. on the system MVA base.A different convergence check evaluates the maximum change in any bus voltage from oneiteration to the next. A solution with desired accuracy is assumed when the change is less thana specified small value, for example, 0.0001 p.u.A voltage check is dependent on the rate of convergence and is thus less reliable than thepower mismatch check. However, the voltage check is much faster (computationally, on adigital computer) than the power mismatch check and since the power mismatch will be large

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